Download - Identification of validated multibody vehicle models …Identification of validated multibody vehicle models for crash analysis using a hybrid optimization procedure Fig. 1 Generic

Struct Multidisc OptimDOI 10.1007/s00158-010-0590-y

INDUSTRIAL APPLICATION

Identification of validated multibody vehiclemodels for crash analysis using a hybridoptimization procedure

Marta Carvalho · Jorge Ambrósio · Peter Eberhard

Received: 29 May 2010 / Revised: 29 August 2010 / Accepted: 22 October 2010c© Springer-Verlag 2010

Abstract The design of components, or particular safetydevices, for road vehicles often requires that many analy-ses are performed to appraise different solutions. Multibodyapproaches for structural crashworthiness provide alterna-tive methodologies to the use of detailed finite elementmodels reducing calculation time by several orders of mag-nitude while providing results with the necessary quality.The drawbacks on the use of multibody approaches forcrashworthiness are the cumbersome model developmentprocess and difficulty of model validation. This work pro-poses an optimization procedure to assist multibody vehiclemodel development and validation. First the topologicalstructure of the multibody system is devised representingthe structural vehicle components and describing the mostrelevant mechanisms of deformation. The uncertainty in themodel development resides on the constitutive behavior ofthe plastic hinges used to represent the structural deforma-tions of the vehicle, identified by using the classic multi-body approach. The constitutive relations are identified byusing an optimization procedure based on the minimiza-tion of the deviation between the observed response of thevehicle model and a reference response, obtained by experi-mental testing or using more detailed models of the vehicle.

M. Carvalho · J. Ambrósio (B)Institute of Mechanical Engineering, Instituto Superior Técnico,Technical University of Lisbon, Lisbon, Portugale-mail: [email protected]

P. EberhardInstitute of Engineering and Computational Mechanics,University of Stuttgart, Stuttgart, Germany

The identification is supported by a genetic optimizationalgorithm that enables the characterization of several initialsets of parameters that characterize the plastic hinge consti-tutive relations in the vicinity of an Edgeworth-Pareto front.Afterwards, a deterministic optimization algorithm is usedto identify the accurate minimum. The procedure is demon-strated by the identification of the multibody model of alarge family car suitable for front impact.

Keywords Multibody models · Multicriteria optimization ·Model validation · Crashworthiness

1 Introduction

In vehicle modeling, details concerning the design are ofmajor importance. However, such aspects often should bekept confidential to external developers, either because theyrepresent technological advances or due to legal reasonsassociated to liability or, therefore, they cannot be dis-closed even to partners. A solution to this problem is thedevelopment of virtual vehicle models that have dynamicresponses similar to the original ones for selected crashscenarios (Sousa et al. 2008). Such responses can be mea-sured in terms of accelerations of given points in the vehiclestructure, energy absorption characteristics of subsystems,intrusion measures or by other measurable characteristics.Alternative approaches to the use of detailed finite elementmodels, which reduce calculation time while providingmeaningful results, are of upmost importance. For a typi-cal frontal crash simulation, with the duration of 150 ms,a finite element model with the appropriate level of detailrequires about 3 days of computational time in a cur-rent desktop computer while a multibody model able todescribe the same mechanisms of deformation, kinematics

M. Carvalho et al.

of the vehicle, contact forces and energy deformation lev-els requires about 3 min. Similar ratios in computation timebetween finite element and multibody models are observedfor other types of crash scenarios. These alternative modelsare developed to be used at the conceptual design develop-ment stage for quick analysis (Kim et al. 2001). A typicalusage of this virtual model would allow a developer to tacklethe task of devising a protective system for a selected part ofthe vehicle being assured that the overall behavior of such avehicle is reasonable, even without being a detailed matchof the original (Sousa et al. 2008).

Advances in computer tools support the design of vehiclestructures for crashworthiness. However, in industrial appli-cations trial and error procedures are still more popular thanthe systematic use of the optimization procedures, espe-cially for highly nonlinear problems, such as those presentin passive safety applications. Structural optimization forcrashworthiness considerations in the automotive industriesare mostly based on finite element models. Bennett et al.(1997) present a methodology for the identification of thefront-end structural stiffness for barrier impact standards.The constitutive relations of the structural components areused in the integration of the equations of motion, beingcoupled to algorithms of optimization in order to improvethe designs obtained by the trial and error procedure. Adrawback of this work was that the component dimen-sions could not be directly manipulated during optimization.Later, Song (1986) used box-beam elements in the numeri-cal model and included a new optimization capability, whichdetermines the dimensions of the structural components tominimize the structural mass while meeting given safety cri-teria. Lust (1992) developed an optimal structural designapproach considering two criteria, simultaneously, associ-ated to the elastic loads and crashworthiness constraints.In order to reduce the computational effort, the crashwor-thiness constraints are nonlinear approximations. Dias andPereira (1994) presented some of the first applications ofoptimal design for crashworthiness using multibody mod-eling approaches. They presented the optimization problemby modeling the vehicle and restraint system by means of asimplified mechanical system. Another approach used byBennett and Park (1995) starts from a detailed and com-plex finite element model for crash and uses an optimizationstrategy, based on the sequential linear programming withsomewhat large finite-difference perturbations, to calculatethe sensitivities. Etman et al. (1996) developed a designoptimization tool for road vehicle crashworthiness that isfirst demonstrated for an analytical test problem with animpact absorber, and applied afterwards to an industrialcrashworthiness design problem consisting in the develop-ment of a restraint system for frontal impact. The crash-worthiness optimization problem is solved by Etman et al.(1996) using sequential approximate optimization in order

to deal with the noisy functional behavior of the objec-tive function and with the high computational costs of theevaluation of the objective functions and constraints. Thesequences of linear programming problems used in theapproach are generated by means of multipoint approxima-tions. Following their earlier work, a design methodologyfor crash structures is presented by Ambrósio et al. (1996)and Dias and Pereira (2004). Complex vehicle structures,such as train sets, are modeled using multibody dynamicsformulations and simulated in crash events. The design andsimulation of these models are linked to deterministic andevolutionary optimization algorithms, which are used in theconceptual design phase to obtain the best characteristics ofthe crash structures and energy absorption devices of trainstructures. The design of railway crash tests with multibodymodels of railway vehicles and deterministic optimizationalgorithms has been proposed by Milho et al. (2004). In allcases of application of optimization methodologies to sup-port the vehicle design for passive safety, the major issueis the identification of proper objective functions and con-straints, which are problem dependent. Also, as far as theoptimization method is concerned, there is no clear favoritewhich one to use.

This work attempts at not only proposing a solution forthe identification of reliable vehicle multibody models forcrashworthiness, but also at identifying, from our experi-ence, a numerically robust optimization procedure to handlethis type of problems with the objective of devising prop-erly validated vehicle models. The input data required forthe model validation is limited to the accelerations, veloci-ties and/or intrusions of the reference vehicle for the crashtest configurations in which the vehicle model is to be used(Seiffert and Wech 2003). It is assumed that the initial modelof the vehicle is obtained using standard good modeling pro-cedures (Nikravesh et al. 1983; Ambrósio 2001; Sousa et al.2008) and that the validated vehicle model is to be used onlyin the crash scenario configurations for which it is validated.

2 Multibody equations

The multibody models are a collection of rigid and/orflexible bodies connected by kinematic joints that restraintheir relative motion. Several types of loads are applied,including spring and damper forces, contact forces result-ing from the impact with the barriers and from the rela-tive motion between the bodies, inertia loads and gravity(Nikravesh 1988). A multibody system can be representedschematically as shown in Fig. 1.

The most commonly used sets of coordinates to describethe motion of each body of a multibody system are Carte-sian, natural and joint coordinates, which are applieddepending on the type of application, complexity of the

Identification of validated multibody vehicle models for crash analysis using a hybrid optimization procedure

Fig. 1 Generic multibody system

multibody system, preference of the research group inwhich the applications are being developed, or, as in thecase of this work, the commercial code which is used. Theefficiency of the numeric methods selected to solve theequations depends on the complexity of the system and onthe set of coordinates chosen (see e.g., Nikravesh 1988;Haug 1989; Schiehlen 1990; Jálon and Bayo 1994).

When the position and orientation of each rigid body isdescribed by Cartesian coordinates, the kinematics relationsmust be described by algebraic constraints. The equationsof motion for the system of nb unconstrained bodies arerepresented by

M q̈ = g (1)

where M is the mass matrix, which includes the masses andinertia tensors of the individual bodies, q̈ is the accelerationvector and g the vector with applied forces and gyroscopicterms. The relative motions between the bodies are con-strained by kinematic joints, which can be mathematicallydescribed by a set of nc algebraic equations, written as

�(q, t) = 0 (2)

The first and second time derivatives of (2) constitute thevelocity and acceleration constraint equations,

�̇(q, t) ≡ �q q̇ − ν = 0 (3)

�̈(q, q̇, t) ≡ �q q̈ − γ = 0 (4)

where λ is a vector with nc unknown Lagrange multipliers, i.e.

M q̈ = g − �Tq λ (5)

Equation (5) has 6nb+ nc unknowns that must be solved to-gether with the second time derivative of the constraint (4).The resulting system of differential-algebraic equations is

[M �T

q

�q 0

] [q̈λ

]=

[gγ

](6)

The stabilization or corrections of the kinematic con-straint drift observed during the integration of (6) can beobtained, e.g., by using the Baumgarte stabilization, theAugmented Lagrangean formulation, or the Coordinate Par-tioning method (Baumgarte 1972; Wehage and Haug 1982;Bayo and Avello 1994; Shabana and Sany 2001; Neto andAmbrósio 2003).

3 Plastic hinge approach to multibody vehicle modeling

The multibody vehicle model for passive safety analysis isbased on the use of the plastic hinge approach (Nikraveshet al. 1983; Ambrósio 2001). This modeling strategyis based on the observation that the structural compo-nents experience plastic deformations in localized areaswhen overloaded. These deformations, presented in Fig. 2,develop at points where maximum bending moments occur,where loads are applied, in structural joints or in locallyweak areas. Some of the first applications of the plastichinge approach are proposed by Nikravesh et al. (1983).Later, Pereira et al. (1987) presented the application to

Fig. 2 Plastic hinge bendingmoment applied in positive (a)and negative (b) axis and itsconstitutive relationship

(a) (b)-10000

-8000

-6000

-4000

-2000

0

2000

4000

6000

8000

10000

-1.5 -1 -0.5 0 0.5 1 1.5Rotation (rad)

Mo

men

t (N

m)

FE MODELMB MODEL

Joint ID 3 Top View

X2

3 1

2

3 1

2

3 1

2

3 1

M. Carvalho et al.

Fig. 3 Plastic hinge models fordifferent load joints

vehicle rollover. The application of the plastic hingeapproach to railway vehicle design has been proposed byMilho et al. (2004). Plastic hinges are now available inmultibody codes such as MADYMO (2004). A furtherextension on this modeling strategy in crashworthinessdesign has been investigated by Pedersen (2003) who pro-posed a finite element with the ability to develop plastichinges on its nodes. The methodology briefly describedherein is also known in the automotive, naval and aerospaceindustries as conceptual modeling (Sousa et al. 2008). Theplastic hinge is modeled by a kinematic joint, describ-ing the kinematics of the deformation, and a generalized

spring-damper element, which is used to represent the con-stitutive characteristics of the elastic-plastic deformation ofthe member.

The characteristics of the spring-damper that describesthe properties of the plastic hinge are obtained by experi-mental component testing, localized finite element nonlin-ear analysis or simplified analytical methods. For a flexuralplastic hinge the spring stiffness is expressed as a functionof the change of the relative angle between two adjacentbodies connected by the plastic hinge as shown in Fig. 2.

This spring-damper is associated with the common typeof joints, applied in multibody modeling as depicted in

Fig. 4 Side structure of avehicle modeled using theplastic hinge approach


Fig. 5 Accelerometers positionin GCM3 model: (a) FE model;(b) MB model

Front_SsillAP

BP_MiddleBP_Bottom

FW_Center

Front_Ssill Front_SsillAP

BP_MiddleBP_Bottom

FW_Center

AP

BP_Middle

BP_Bottom

FW_Center

AP

BP_Bottom

FW_Center

(a) (b)

Fig. 3, for one axis bending, two axis bending, torsion andaxial, forming what is defined as plastic hinge (Ambrósio2001). Then, a structural component can be modeled as acollection of rigid bodies connected by plastic hinges, asshown in Fig. 4 for the side structure of a road vehicle.By implementing a proper discretization of the structureit is possible to model its most relevant mechanisms ofdeformation, i.e., not only each structural component of thevehicle must be modeled by a correct number of rigid bod-ies and plastic hinges according to good modeling practices(Nikravesh et al. 1983; Ambrósio 2001; Sousa et al. 2008)but also the model has to be able to develop the mechanismsof deformation characteristics of the crash configurations inwhich it is used.

Note that although the approach is deemed as ‘plastic’hinge, the deformations observed may also be elastic only.Although the formulation implied here only considers rigid

bodies connected by kinematic joints of the plastic hinges,the methodology can be extended to include flexible bodies(Dias and Pereira 1994).

4 Multibody models for vehicles

The generic car model developed in this work is to be usedin crash test scenarios described by the regulations approvedin Europe. Of particular relevance are the ECE regulationsfor frontal and side impact that must be fulfilled by any newvehicle that seeks approval for release in the EU countries.In the application described here the GCM3 model is testedaccording to the ECE R33 European Regulation (1995) forfull frontal rigid barrier impact. Due to the impossibility toaccess the experimental data a surrogate FE model testedaccording to the ECE R33 regulation, validated for frontal

Fig. 6 Deformations of GCM3MB model developed withnon-optimal plastic hingeconstitutive relations andthe FE model for the ECE R33crash test 0 ms

20 ms

40 ms

60 ms

M. Carvalho et al.

Fig. 7 Dynamic responsemeasured at the A-Pillar for theMB model obtained aftermodeling structural deformationwith plastic hinge approach

impact as reported in (Puppini et al. 2005), is used to supplythe data required to the construction and validation of theMB model. Note that real vehicle data can be used directlyas reference data with the proposed procedure when avail-able. The frontal crash test is conducted with the GCM3 FEmodel for the regulatory impact speed of 48.3 km/h.

The crash response of the vehicle can be measured interms of accelerations, velocities and intrusions in selectedpoints of the structure such as those shown in Fig. 5. It iscommon practice to filter the acceleration signals with the

CFC 60 filter and the velocities and displacements with CFC180 filter, for both FE and MB models.

The deformations of the finite element reference modeland of the multibody vehicle model for ECE R33 crash sce-nario are shown in Fig. 6 for a sequence of instants in time.It is observed that although the mechanisms of deformationare similar, the multibody model shows here a much stifferappearance than that of the finite element model.

In order to quantify the outcome of the dynamic anal-ysis of the vehicle model obtained with the recommended

Fig. 8 Dynamic responsemeasured at the middle B-Pillarfor the MB model obtained aftermodeling structural deformationwith plastic hinge approach


multibody modelling approaches, the accelerations, veloc-ities and displacements in the longitudinal direction forthe pick-up points located in A-Pillar and middle B-Pillar,are presented in Figs. 7 and 8, respectively. These crashresponses show that the dynamic responses of the multibodyvehicle model are not yet well correlated with the referenceresponse of the reference vehicle. Such lack of correlationis also observed in all other measured crash responses notshown here. In an industrial environment the analyst manu-ally tunes the model parameters associated with the plastichinges until its response matches, or is sufficiently close,that of the reference. This procedure, besides being far fromoptimal, relies completely in the experience of the analyst.Therefore, a systematic approach to the calibration of themodel by using optimization is introduced in this work.

The model already contains the mechanisms of defor-mation of the vehicle, which is a fundamental featurefor the success of what follows. The real deformation ofeach member of the structural setup includes the gener-alized elastic deformation of the member plus the plasticdeformation of the plastic hinge region. The plastic hingeapproach disregards the generalized elastic deformation ofthe structural members and considers the plastic deforma-tion region as lumped. Two corrective measures can beforeseen for the MB models; one is to increase the num-ber of plastic hinges in each structural member in orderto better distribute the deformation and, the second is tomodify some plastic hinges constitutive functions so thatthe generalized elastic deformation that develops in theirneighbouring regions can be accounted for. The use of thefirst corrective measure implies changing the topology ofthe multibody model by modifying the number of rigid bod-ies and kinematic joints used in the model. This change ofthe model structure must always be done by manually withgreat care. As no automatic procedure exists in the currentmultibody methodology to modify the level of discretizationof the structural elements a suitable computational proce-dure should be devised first. This is no trivial task, sincein multibody systems no mathematically well-defined errormeasures are known. Therefore, in this work the secondcorrective measure, in which the plastic hinge constitutiverelations are modified by using optimization approaches, isselected.

5 Identification of the multibody vehicle modelusing optimization methods

The model adaptation is the next step in the model devel-opment and constitutes in fact an optimization process toidentify the MB model that best represents the vehicle. Theobjective of the identification problem is to achieve a MBmodel that has displacements, velocities and accelerations

that correlate well with those measured in the same struc-tural locations in a reference vehicle. In what follows theidentification of the vehicle model is pursued only for thefront crash impact scenario. Although a MB vehicle modelsuitable for several crash impact scenarios can be devisedusing the proposed methodology, front and side impact forinstance (Carvalho and Ambrosio 2010), the idea of a modelsuitable for omni-directional impacts, if achievable, wouldbe a complex task.

5.1 Design variables

At this stage, the parameter values in the constitutive equa-tions and the locations of the plastic hinges are adjustedwithin a given range of variation, to increase the correla-tion between the behavior of the reference and multibodymodels. This has been the procedure used (Mooi et al.1999; Gielen et al. 2000; van der Zweep and Lemmen2002; van der Zweep et al. 2005). The acceptable rangeof variation of the model data is related to the approxima-tions made when defining the discretization of the modeland the constitutive equations of the plastic hinges. Theseranges are represented as corridors inside which differentforce-displacement behaviors are accepted, see e.g. Fig. 9.

For the MB model, the plastic hinges are identifiedtogether with their variation ranges. The locations of theplastic hinges that supply the design variables for the opti-mization are represented for the left side of the car inFig. 10. To demonstrate the procedure a simplified situationwith just five design parameters is used in what follows. Dueto the ability to change the model crash behavior by scal-ing the plastic hinge constitutive relation (Sousa et al. 2008)and to the complexity of the vehicle multibody models

Angle

Moment

Fig. 9 Constitutive relation for a generic plastic hinge and variationcorridor

M. Carvalho et al.

Fig. 10 Joint localization and respective design variables for theGCM3-MB model

(Ambrósio and Dias 2007), the design variables x used inthis work are the scaling factors of selected constitutivefunctions:

1. Forces on the plastic hinge of the body structure,2. Forces on the plastic hinges of the frame,3. Forces on the plastic hinges of the bumper frame,4. Forces on the plastic hinges of the sub-frame,5. Forces on the plastic hinges of the bumper sub-frame.

It must be noticed that each of the parameters selectedas design variable affects several plastic hinges that sharethe same constitutive relation. Therefore, each parameterinfluences the structural behavior of parts of substructuresof the vehicle model.

5.2 Optimization criteria

In this work the time interval of the analysis is discretizedinto time points and the objective functions are evaluated asthe sum of the square deviation of the vehicle responses insuch instants, as implied by Fig. 11. For each of the dynamicresponses selected, the error between the reference and the

Ay

Model Response

Reference Response

timet0 t1 t2 t3 t4 t5 t6 t7

Fig. 11 Time responses of the MB and reference models measured ina selected point of the vehicle

MB model response, at a finite number N of instants, is cal-culated as errors of acceleration, fi,acc(xxx), velocity, fi,vel(xxx)

and intrusion, fi,disp(xxx). These errors are defined as

fi,acc(xxx) =N∑

j=1

√(acci

MB(xxx, t j ) − accire f (t j )

)2 (7)

fi,vel(xxx) =N∑

j=1

√(veliMB(xxx, t j ) − velire f (t j ))2 (8)

fi,disp(xxx) =N∑

j=1

√(dispi

MB(xxx, t j ) − dispire f (t j )

)2 (9)

Due to the potentially conflicting responses or due to theimpossibility of minimization of all responses simultane-ously an appropriate trade-off between objectives has to befound. Usually these problems do not have a unique solu-tion. The concept of Edgeworth-Pareto (EP) optimality canbe used to characterize the objectives (Censor 1977; Cunhaand Polak 1967; Deb 2001).

The purpose of a multicriteria optimization problem isdefined as finding the design variables xxx that

Optimize F(xxx) n-vector of objective functions

subject to gi (xxx) ≤ 0 inequality constraints,

i = 1, . . ., m

hj (xxx) = 0 equality constraints,

j = 1, . . ., p

xL ≤ x ≤ xU bounds

Instead of a single minimum of a scalar optimizationproblem, the multiobjective optimization problem leads toa set of EP-optimal designs, also called EP front. In mul-ticriteria problems there are several typical approaches: thefirst is to find a solution on the EP front, and the second is tofind a set of solutions as diverse as possible. Mathematicallyit is not possible to prefer one EP solution over another with-out further user-specified information. If this information isavailable, it can be used to make a choice. In the absence ofsuch information, EP-optimal solutions are not comparable(Deb 2001).

Genetic algorithms do not require gradient informationand that they also allow reaching an EP front (Deb 2001).Their disadvantage is the high number of function evalua-tions required. Therefore, several generations of the geneticalgorithm are used in this work to find sets of starting designvariables that are close to the EP front. Then, these designsare used as starting points for a gradient based optimiza-tion method to obtain efficiently minima of the problem.


The best solution among all minima obtained is deemed asthe solution of the optimization problem. This is a hybridapproach that takes advantage of the ability of the geneticalgorithms to find suitable starting designs and the suitabil-ity of the gradient based algorithms to converge to minimafrom each one of those designs.

5.3 Formulation of the optimization problem

In this work the objective is to identify a vehicle multi-body model that has prescribed crash responses. First, theproblem can be defined as a multiobjective optimizationfor the search of EP front (Ambrósio and Eberhard 2009).Afterwards the model identification is reduced to a scalaroptimization problem

minx

F(xxx) =n∑

i=1

N∑j=1

√(acci

M B(xxx, t j ) − accire f (t j )

)210−4

(10)

subject to

1

N

N∑j=1

√(veliM B(xxx, t j )−velire f (t j )

)2 ≤ f ∗i,vel

, i = 1, . . . , n

(11)

0.1 ≤ xi ≤ 2, i = 1 to 5

Table 1 Values for the scalar objective function and constraints foreach design of the MB model

Design F(xxx)N∑

j=1

√(veliMB(xxx, t j ) − velire f (t j ))2

A-pillar Fire wall Front B-pillar B-pillar

center sill (middle) (bottom)

1 3.57 38 47 43 57 41

2 5.48 46 72 46 64 44

3 5.31 72 91 70 98 71

4 4.17 43 43 49 47 47

5 4.36 45 65 47 74 46

6 5.60 54 81 47 88 49

7 4.24 42 46 47 52 45

8 4.84 81 96 78 115 80

9 4.64 62 78 54 99 57

where the objective function is defined as the sum of themean square errors of the n accelerations measured indifferent points of the vehicle, represented in Fig. 5, andsampled on N instants in time for the duration of the crashevent, as represented in Fig. 11. In the case of this studyn = 5 and N = 60, which means a time sampling of 1 ms.

It is assumed here that all acceleration criteria are equallyimportant. The errors in the intrusion velocity profiles of thevehicle are used as constraints. The summation of the meansquare deviation between the model and the goal velocitiesfor each sensor is depicted by f ∗

i,vel.

Fig. 12 Deformation of thereference and multibody vehiclemodels 0 ms

20 ms

40 ms

60 ms

M. Carvalho et al.

6 Results of the application: validated multibodymodels

In this implementation, the genetic algorithm is used in firstplace to identify the initial designs that are later exploredby using a Sequential Quadratic Programming (SQP) algo-rithm. The genetic algorithm stops after a preset numberof generations, which are deemed enough to characterizea group of individuals that constitute the EP front. Deb’selitist algorithm NSGA-II is the selected genetic algorithmused in this work (Deb et al. 2002). This algorithm is imple-mented in the function ‘gamultiobj’ available in MATLABGenetic Algorithm Toolbox (MATLAB 2008). This geneticalgorithm is part of the MOEA (Multiobjective Evolution-ary Algorithm) family, which have the ability to handlecomplex problems involving features such as discontinu-ities, multimodality, disjoint feasible spaces and noisy func-tions evaluations, reinforcing their potential effectivenessin multiobjective search and optimization (Fonseca andFleming 1995; Eberhard et al. 2003).

The number of sets of design variables used here is 15times the number of design variables, i.e., 75 in the opti-mization problem proposed in this work. The initial designis randomly created in the feasible region. In each gen-eration a selection function chooses parents for the nextgeneration, based on their scaled values from the fitnessfunctions in which a ‘tournament’ is used. Each parent isselected by choosing two individuals at random, and thenselecting the best individual among them. Reproductionis done by scattered crossover with a rate of 95%, being1% of mutation rate introduced to provide genetic diver-sity and to enable the genetic algorithm to search a broader

Table 2 Measure of the criterion for the initial design and optimizedMB model

Scalar obj. fi,acc(×104) for accelerometer i

function A-pillar Fire wall Front B-pillar B-pillarF(xxx) center sill (middle) (bottom)

Initial 4.36 0.69 2.00 0.61 0.43 0.63

Optimal 3.26 0.58 0.96 0.60 0.54 0.58

space. The Edgeworth-Pareto front population fraction isanother parameter that keeps the fittest population downto the specified fraction, which is 35% in this application,to maintain a diverse population. In this implementation,the genetic algorithm is used to provide the initial designsfor SQP algorithm. The number of generations in this caseis set to 10. The number of stall generations is set to 1.The improvement of the objective function, which is theweighted average change in fitness value over stall gener-ations, is measured. If this value is less than a functiontolerance, the algorithm stops.

The first nine candidates for the EP front are computedhere after the calculation of three generations in 7.5 h ofCPU time. Each crash analysis of the vehicle MB modeltakes about 3 min of CPU time. The values of the scalarobjective function, described by (10), and velocity errorconstraints are listed in Table 1. The locations of the sen-sors used to formulate the objective functions, referred to inTable 1, are listed in Fig. 5.

Instead of continuing to generate more designs by con-sidering more generations, which is a time consuming pro-cedure, the next step is to continue with an SQP algorithm.

Fig. 13 Dynamic responsemeasured at the A-Pillar for theMB models obtained after GAand SQP versus the FE modelfor the front impact test


Fig. 14 Dynamic responsemeasured at the middle B-Pillarfor the MB models obtainedafter GA and SQP versus the FEmodel for the front impact test

The initial designs used for each analysis with deterministicalgorithm are the already computed intermediate solutions.The SQP optimization algorithm is available in the MAT-LAB Optimization Toolbox (MATLAB 2008) as function‘fmincon’, based on a sequential programming approachwith active-set strategy.

The F(xxxopt ) = 3.26 design is achieved in the 5th itera-tion for the vector of design variables xxxopt = [0.10, 0.25,

0.10, 0.10, 0.10]. The patterns of deformation of the refer-ence and optimized vehicle multibody models are depictedin Fig. 12. It is observed that the major characteristics of thedeformation are now well correlated. This qualitative infor-mation gives an impression of the improved behavior of thevalidity of the model but it is not enough to deem it as beingvalidated.

The errors on the dynamic response for the initial designand for the optimized MB model are summarized in Table 2.The results show an improvement of almost all dynamicresponses, except accelerations on the middle of B-pillar, asseen in Table 2 and illustrated by Figs. 13 and 14 that presentselected dynamic responses of the reference model and thebest MB model obtained using the gradient based algorithmand the initial model defined by the 5th design identifiedwith the genetic algorithm. The vehicle MB model obtainedwith the hybrid algorithm shows a good correlation withthe reference model and all acceleration responses are bet-ter correlated with the reference responses than those of theinitial model.

The hybrid algorithm approach applied to the problemof validating the multibody vehicle model for front impacttakes advantage of the diversity and potential location for

the minima with the genetic algorithm in few generations,and after that applies the SQP method to get faster con-vergence from there. The genetic algorithm takes 7.5 hof computational time to obtain designs suitable for thenext step of optimization undertaken by the SQP algorithm.Based on good initial designs obtained with the geneticalgorithm, the convergence by the SQP method is reached inonly five iterations, taking about 1 h of additional CPU time.Based on the methodology described in this work moredesign variables and criteria can be easily accommodated,leading to better vehicle models.

7 Conclusions

A methodology for the construction of validated multibodymodels of vehicles for impact analysis has been demon-strated in this work, being the optimal model identifiedby the hybrid approach proposed. First the genetic opti-mization algorithm generates a group of designs usuallyclose to the EP front. The use of gradient based algorithm,in this case the SQP, explores each one of these designsallowing to find the best of them, deemed here as the val-idated vehicle multibody model. It has been observed thatthe identification of the design variables as proportionalfactors of the plastic hinge constitutive relations is suit-able not only because they represent what is uncertain inthe construction of multibody models for crash but alsobecause they allow obtaining a vehicle model with good pre-dictability. Although the objective function can be expressedby intrusion, velocity and acceleration errors between the

M. Carvalho et al.

reference and multibody models dynamic responses, it hasbeen shown that in the case of frontal impact by usingthe acceleration errors as the objective functions and thevelocity errors as constraints leads to a robust optimizationprocess with results that show a good correlation betweenmodels while improving the convergence of the optimiza-tion problem. The drawback of this approach is that theconstraints of the problem become nonlinear. The method-ology was demonstrated in the development of a multibodymodel for a large family car for which a validated finiteelement model exists and is used as reference. Becausethe approach only uses the dynamic response of the vehi-cle that can be obtained in a crash test, the methodologyproposed can be applied to the development of multibodymodels when only generic models exist and the exact mod-els are unavailable for the user. The MB vehicle modelpresented in this work is suitable for a front crash scenarioonly. However, by using reference data for other scenarioconfigurations it is possible to obtain a vehicle model suit-able for multi-directional impact or to obtain specializedmodels valid for multiple crash scenarios. The two impor-tant issues to be taken into account when using the validatedmodel are that in can only used in the crash scenario inwhich it has been validated and that no modifications to themodel of the structural components responsible for its crash-worthiness behavior can be done without further validation.Under these conditions, its use for the development of safetysystems such as airbags, seats and any other protectivedevice is possible without further need for validation.

Acknowledgments The support of EU through project APROSYS-SP7 with the contract number TIP3-CT-2004-506503, having aspartners Mecalog (F), CRF (I), TNO (NL), Politecnico di Torino(I), CIDAUT (ES), Technical University of Graz (A) is gratefullyacknowledged. The financial support of the first author received fromFCT—Fundação para a Ciência e Tecnologia through grant SFRH/BD/23296/2005 is highly appreciated.

References

Ambrósio J (2001) Crashworthiness: energy management and occu-pant protection. Springer, Wien

Ambrósio J, Dias J (2007) A road vehicle multibody model for crashsimulation based on the plastic hinges approach to structuraldeformations. Int J Crashworthiness 12(1):77–92

Ambrósio J, Eberhard P (2009) Advanced design of mechanicalsystems: from analysis to optimization. Springer, Wien

Ambrósio J, Pereira MS, Dias J (1996) Distributed and discretenonlinear deformations in multibody dynamics. Nonlinear Dyn10(4):359–379

Baumgarte J (1972) Stabilization of constraints and integrals ofmotion. Comput Methods Appl Mech Eng 1:1–16

Bayo E, Avello A (1994) Singularity-free augmented lagrangianalgorithms for constrained multibody dynamics. Nonlinear Dyn5:209–231

Bennett JA, Park GJ (1995) Automotive occupant dynamics optimiza-tion. Shock Vib 2:471–479

Bennett JA, Lin KH, Nelson MF (1997) The Application of optimiza-tion tecniques to problems of automotive crashworthiness. SAEpaper No. 770608:2255–2262

Carvalho M, Ambrosio J (2010) Identification of multibody vehiclemodels for crash analysis using an optimization methodology.Multibody Syst Dyn 24(3):325–345

Censor Y (1977) Pareto optimality in multiobjective problems. ApplMath Optim 4:41–59

Cunha NOD, Polak E (1967) Constrained minimization under vectorvalued criteria in finite dimensional spaces. J Math Anal Appl19:103–124

Deb K (2001) Multi-objective optimization using evolutionary algo-rithms. Wiley, Chichester

Deb K, Agrawal S, Pratap A, Meyarivan T (2002) A fast eli-tist non-dominated sorting genetic algorithm for multi-objectiveoptimization: NSGA-II. IEEE Trans Evol Comput 7(2):182–197

Dias JP, Pereira MS (1994) Design for crashworthiness using multi-body dynamics. Int J Veh Des 15(6):563–577

Dias JP, Pereira MS (2004) Optimization methods for crashworthi-ness design using multibody models. Comput Struct 82:1371–1380

Eberhard P, Dignath F, Kübler L (2003) Parallel evolutionary optimiza-tion of multibody systems with application to railway dynamics.Multibody Syst Dyn 9(2):143–164

Etman LFP, Adriaens JMTA, van Slagmaat MTP, Schoofs AJG (1996)Crash worthiness design optimization using multipoint sequentiallinear programming. Struct Optim 12:222–228

European Council for Europe (1995) Regulation 33—uniform pro-visions concerning the approval of vehicles with regard to thebehaviour of the structure of the impacted vehicle in a head-oncollision

Fonseca CM, Fleming PJ (1995) An overview of evolutionary algo-rithms in multiobjective optimization. Evol Comput 3(1):1–16

Gielen AWJ, Mooi HG, Huibers JHAM (2000) An optimizationmethodology for improving car-to-car compatibility. IMechETransactions C567/047/2000

Haug E (1989) Computer aided kinematics and dynamics of mechani-cal systems. Allyn and Bacon, Boston

Jálon J, Bayo E (1994) Kinematic and dynamic simulation of multi-body systems: the real-time challenge. Springer, New York

Kim CH, Mijar AR, Arora JS (2001) Development of simplifiedmodels for design optimization of automotive structures for crash-worthiness. Struct Multidisc Optim 22:307–321

Lust R (1992) Structural optimization with crashworthiness con-straints. Struct Optim 4(2):85–89

MADYMO version 6.2 (2004) Theory manual. TNO MADYMO BV,Delft, Netherlands

MATLAB (2008) Release notes. The Mathworks, Inc., Natick,Massachusetts

Milho J, Ambrósio J, Pereira M (2004) Design of train crash exper-imental tests by optimization procedures. Int J Crashworthiness9(5):483–493

Mooi HG, Nastic T, Huibers JHAM (1999) Modelling and optimiza-tion of car-to-car compability. VDI Berichte NR 1471:239–255,Dusseldorf, Germany

Neto MA, Ambrósio J (2003) Stabilization methods for the integrationof DAE in the presence of redundant constraints. Multibody SystDyn 10:81–105

Nikravesh P (1988) Computer-aided analysis of mechanical systems.Prentice-Hall, Englewood Cliffs

Nikravesh PE, Chung IS, Benedict RL (1983) Plastic hinge approachto vehicle crash simulation. Comput Struct 16(1–4):395–400

Pedersen CBW (2003) Topology optimization design of crushed 2D-frames for desired energy absorption history. Struct MultidiscOptim 25:368–382


Pereira MS, Nikravesh P, Gim G, Ambrósio J (1987) Dynamic analysisof roll-over and impact of vehicles. XVIII Bus and Coach ExpertsMeeting, Budapest, Hungary

Puppini R, Diez M, Ibba A, Avalle M, Ciglaric I, Feist F (2005)Generic car (FE) models for categories super minis, small familycars, large family executive cars, MPV and heavy vehicle. Tech-nical Report APROSYS SP7-Virtual Testing, AP-SP7-0029-A

Schiehlen W (1990) Multibody systems handbook. Springer, BerlinSeiffert U, Wech L (2003) Automotive safety handbook. SAE Interna-

tional, WarrendaleShabana AA, Sany JR (2001) An augmented formulation for mechan-

ical systems with non-generalized coordinates: an applicationto rigid body contact problems. Nonlinear Dyn 24(2):183–204

Song JO (1986) An optimization method for crashworthiness design.SAE Transactions, SAE Paper No. 860804:39–46

Sousa L, Veríssimo P, Ambrósio J (2008) Development of genericroad vehicle models for crashworthiness. Multibody Syst Dyn19(1):135–158

van der Zweep CD, Lemmen PL (2002) Improvement of crash com-patibility between passenger cars by numerical optimization offront-end stiffness. SAE Transactions, SAE Paper No. 5353

van der Zweep CD, Kellendonk G, Lemmen P (2005) Evaluation offleet systems model for vehicle compatibility. Int J Crashworthi-ness 10(5):483–494

Wehage RA, Haug EJ (1982) Generalized coordinate partitioning fordimension reduction in analysis of constrained dynamic systems.J Mech Des 104:247–255